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Sharp Non-uniqueness in Law for Stochastic Differential Equations on the Whole Space

Huaxiang Lü, Michael Röckner

TL;DR

The paper proves sharp non-uniqueness in law for stochastic differential equations on $\mathbb{R}^d$ with divergence-free drift in the supercritical regime by constructing two distinct positive density evolutions via a novel convex integration scheme extended to the whole space. Central to the approach is decomposing the stress into a principal compact part and a small global remainder, and introducing $L^m$-based intermittent jets to reach $\frac{d}{p}+\frac{1}{r}>1$; heat-kernel estimates anchor the positivity and speed of propagation. The result yields non-uniqueness for multiple initial measures and, via the superposition principle, for a set of initial points with positive Lebesgue measure. This advances understanding of weak well- and ill-posedness for SDEs with rough drifts in the critical-to-supercritical regime and demonstrates the power of convex integration methods in non-periodic, unbounded domains.

Abstract

In this paper, we investigate the stochastic differential equation on $\mathbb{R}^d,d\geq2$: \begin{align*} \dif X_t&=v(t,X_t)\dif t+\sqrt{2} \dif W_t. \end{align*} For any finite collection of initial probability measures $\{μ^i_0\}_{1\leq i\leq M}$ on $\mathbb{R}^d$ and $\frac{d}{p}+\frac{1}{r}>1$, we construct a divergence-free drift field $v\in L_t^rL^p\cap C_tL^{d-}$ such that the associated SDE admits at least two distinct weak solutions originating from each initial measure $μ^i_0$. This result is sharp in view of the well-known uniqueness of strong solutions for drifts in $C_tL^{d+}$, as established in \cite{KR05}. As a corollary, there exists a measurable set $A\subset\mathbb{R}^d$ with positive Lebesgue measure such that for any $x\in A$, the SDE with drift $v$ admits at least two weak solutions when with start in $x\in A$. The proof proceeds by constructing two distinct probability solutions to the associated Fokker-Planck equation via a convex integration method adapted to all of $\mathbb{R}^d$ (instead of merely the torus), together with refined heat kernel estimate.

Sharp Non-uniqueness in Law for Stochastic Differential Equations on the Whole Space

TL;DR

The paper proves sharp non-uniqueness in law for stochastic differential equations on with divergence-free drift in the supercritical regime by constructing two distinct positive density evolutions via a novel convex integration scheme extended to the whole space. Central to the approach is decomposing the stress into a principal compact part and a small global remainder, and introducing -based intermittent jets to reach ; heat-kernel estimates anchor the positivity and speed of propagation. The result yields non-uniqueness for multiple initial measures and, via the superposition principle, for a set of initial points with positive Lebesgue measure. This advances understanding of weak well- and ill-posedness for SDEs with rough drifts in the critical-to-supercritical regime and demonstrates the power of convex integration methods in non-periodic, unbounded domains.

Abstract

In this paper, we investigate the stochastic differential equation on : \begin{align*} \dif X_t&=v(t,X_t)\dif t+\sqrt{2} \dif W_t. \end{align*} For any finite collection of initial probability measures on and , we construct a divergence-free drift field such that the associated SDE admits at least two distinct weak solutions originating from each initial measure . This result is sharp in view of the well-known uniqueness of strong solutions for drifts in , as established in \cite{KR05}. As a corollary, there exists a measurable set with positive Lebesgue measure such that for any , the SDE with drift admits at least two weak solutions when with start in . The proof proceeds by constructing two distinct probability solutions to the associated Fokker-Planck equation via a convex integration method adapted to all of (instead of merely the torus), together with refined heat kernel estimate.

Paper Structure

This paper contains 12 sections, 4 theorems, 68 equations.

Table of Contents

  1. Introduction
  2. Main result
  3. Idea of the proof
  4. $L^m$-based intermittent spatial-time jets
  5. Proof of Main result
  6. Estimate on $\rho^{in,i}$
  7. Convex integration set up
  8. Proof of main result
  9. Proof of Proposition \ref{['prop:case4']}
  10. Choice of parameters
  11. Mollification
  12. Construction of $v_{q+1}$ and $\rho^{i}_{q+1}$

Key Result

Theorem 1.1

Let $d\geqslant2,\gamma\in(0,1),1<s<d$ and $p,r\in[1,\infty]$ satisfying $\frac{d}{p}+\frac{1}{r}>1$. For a finite collection of initial distributions $\{\mu^i_0\}_{1\leqslant i\leqslant M}\subset{\mathcal{P}}$, there exists a divergence-free vector field $v\in L^r([0,T];L^p) \cap C([0,T];L^s)$ such

Theorems & Definitions (7)

  • Theorem 1.1
  • Corollary 1.2
  • Definition 1.3
  • Theorem 1.4
  • Lemma 2.1
  • proof : Proof of Theorem \ref{['thm:non_pde_4']}
  • proof : Proof of Theorem \ref{['thm:4converge']}